The perimeter of a certain sector of a circle is equal to the length of the arc of a semi-circle having the same radius ; express the angle of the sector in degrees, minutes and seconds.

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Solution

Let $r$ be the radius of the circle and $θ$ the number of radians in the angle.
Then $r θ + 2 r = π r$ or $θ = (π - 2)$ radians
or $θ = (π - 2) \times \frac{180}{π}$
or $= 180 - 2 \times 5717^{'} {44.8}^{''}$
or $θ = 180 - 11435^{'} {29.6}^{''} = 6524^{'} {30.4}^{''}$

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The perimeter of a certain sector of a circle is equal to the length of the arc of a semi-circle having the same radius ; express the angle of the sector in degrees, minutes and seconds.

Let r be the radius of the circle and θ the number of radians in the angle.Then rθ+2r=πr or θ=(π−2) radiansor θ=(π−2)×180πor =180−2×5717′44.8′′ or θ=180−11435′29.6′′=6524′30.4′′

Let $r$ be the radius of the circle and $θ$ the number of radians in the angle.
Then $r θ + 2 r = π r$ or $θ = (π - 2)$ radians
or $θ = (π - 2) \times \frac{180}{π}$
or $= 180 - 2 \times 5717^{'} {44.8}^{''}$
or $θ = 180 - 11435^{'} {29.6}^{''} = 6524^{'} {30.4}^{''}$